Crease pattern of Mooser's Train removed due to copyright restrictions.

Refer to: Fig. 12.4 from Lang, Robert J. Origami Design Secrets: MathematicalMethods for an Ancient Art. 2nd ed. A K Peters / CRC Press, 2011.

1

folding by Robert Lang

Courtesy of Robert J. Lang. Used with permission.

Crease pattern of Mooser's Train removed due to copyright restrictions.

Refer to: Fig. 12.4 from Lang, Robert J. Origami Design Secrets: MathematicalMethods for an Ancient Art. 2nd ed. A K Peters / CRC Press, 2011.

2

Black Forest Cuckoo Clock, opus 182 Robert Lang, 1987

Courtesy of Robert J. Lang. Used with permission. 3

Are there known universal hinge patterns to build poly-some-other-shapes-that-are-not-cubes?

Folded state of box-please gadget removed due to copyright restrictions.

4

Yes/No

Demaine, Demaine,

Stengle 2011 Courtesy of Erik Demaine, Martin Demaine, and Sarah Stengle. Used with permission.

5

Science/Art Demaine & Demaine, 2011 Courtesy of Erik Demaine, and Martin Demaine. Used with permission.

6

Martin Gardner

Demaine, Demaine

2012 Courtesy of Erik Demaine, and Martin Demaine. Used with permission.

7

GLASS Demaine & Demaine, 2012

Courtesy of Erik Demaine, and Martin Demaine. Used with permission.

8

I didn’t understand the point of NP-hardness. Are there examples of actual problems that can't be calculated?

9

Could we go through one of the NP proofs with a little less hand waving?

10

[Arkin, Bender,

Demaine, Demaine, Mitchell, Sethia, Skiena 2000] Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission. 11

[Arkin, Bender,

Demaine, Demaine, Mitchell, Sethia, Skiena 2000] Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission. 12

Minor question: in the orthogonal paper reduction, doesn’t this require not folding some of the creases, if we want to make 2 consecutive strips the same direction?

13Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission.

[Bern & Hayes 1996]

14

Not All Equal(x1.x2.x3)

x3

x3

x2

x2x1

x1

Image by MIT OpenCourseWare.

[Bern & Hayes 1996]

Not All Equal(x1.x2.x3)

x3

x3

x2

x2x1

x1

15

Image by MIT OpenCourseWare.

In the reflector gadget, it looks like all the left sides of the wires, where left is taken relative to the free end of the wire, are equal. How does the reflector negate one of them, then?

[Bern & Hayes 1996]

16

θTr

ue

True

False

True

False

Image by MIT OpenCourseWare.

It looks like the global flat foldability proof proves that globally flat-foldable ⇒ NAE satisfiability ⇒ locally flat-foldable, but I don't see where NAE satisfiability ⇒ globally flat-foldable. (It looks like all that matters is the order of sheets, though, and that those all work out.)

17

[Bern & Hayes 1996]

18

Not All Equal(x1.x2.x3)

x3

x3

x2

x2x1

x1

Image by MIT OpenCourseWare.

For global flat foldability, I understand how the gadgets prove (1), but how do they prove (2)?

19

[Bern & Hayes 1996]

tab

wire

true false

20

NAE clause

[Bern & Hayes 1996] 21

22

Front Back

[Demaine & Demaine 2003]

Photographs of MIT Stata Center, dome, and Simmons Hall © source unknown. All rights reserved. This content is

excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.23

[Demaine & Demaine 2003]

1,368 folded states

24

[Demaine, Liu, Morgan 2012]

NEWS labeling

Courtesy of Erik Demaine, Eric Liu, and Thomas Morgan. Used with permission.

25

[Demaine, Liu, Morgan 2012]

top edge view

Courtesy of Erik Demaine, Eric Liu, and Thomas Morgan. Used with permission.

26

[Demaine, Liu, Morgan 2012]

top edge view

East West

North South

left right

out in

Courtesy of Erik Demaine, Eric Liu, and Thomas Morgan. Used with permission.

27

[Demaine, Liu, Morgan 2012]

ray diagram East West

North South

down up

nesting depth

left right

out in

East West

North South

Courtesy of Erik Demaine, Eric Liu, and Thomas Morgan. Used with permission.

28

[Demaine, Liu, Morgan 2012]

East West

North South

down up

nesting depth

constrained constrained unconstrained

valid invalid

ray diagram

Courtesy of Erik Demaine, Eric Liu, and Thomas Morgan. Used with permission.

29

[Demaine, Liu, Morgan 2012]

Courtesy of Erik Demaine, Eric Liu, and Thomas Morgan. Used with permission.

30

SSESNSNSWSNWNSN

[Demaine, Liu, Morgan 2012]

Courtesy of Erik Demaine, Eric Liu, and Thomas Morgan. Used with permission.

31

[Demaine, Liu, Morgan 2012]

Courtesy of Erik Demaine, Eric Liu, and Thomas Morgan. Used with permission.

32

MIT OpenCourseWarehttp://ocw.mit.edu

6.849 Geometric Folding Algorithms: Linkages, Origami, PolyhedraFall 2012

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.